Theorem annotanannot 832

Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.)

Assertion

Ref	Expression
annotanannot	⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem annotanannot

Step	Hyp	Ref	Expression
1		ibar 531	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
2	1	bicomd 225	. . 3 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓))
3	2	notbid 320	. 2 ⊢ (𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ ¬ 𝜓))
4	3	pm5.32i 577	1 ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by:  clwwlknclwwlkdif  27757  0nn0m1nnn0  32351